978 twists... Log is here: http://shade.msu.ru/~astr/MC4D/4d4_978.log
I'll try to describe the method later.
Andrey
Very impressive, Andrey! It was just a few days ago that I mentioned to
you that I'd love to see a sub-1000 twist 4^4 solution and today you
give us one. This is certainly a landmark achievement. It feels to me
that we are now seeing something like the true difficulty of n^4
solutions using certain methods. Of course you are using a rather new
approach which I'm sure everyone on this list will be interested to hear
about. I bet that several members will quickly figure out which puzzles
it can be applied to and maybe we'll see a few more broken records
before too long. Thank you for developing new methods and
congratulations on your landmark record.
-Melinda
Andrey wrote:
> 978 twists... Log is here: http://shade.msu.ru/~astr/MC4D/4d4_978.log
> I'll try to describe the method later.
Melinda, Remi, thank you!
I haven't try 2^4 yet. But it seems to be much more 4D than large cubes -=
so it will be interesting to do something with it )))
As for real difficulties of n^4, I think that we are very far from it. Al=
l human algorithms are based on some restricitions of possible actions - so=
we are closing short ways and select long but predictable. I tried to clos=
e as few ways as possible - but to have small enough set of resulting posit=
ions. I'm sure that there better ways (that use full power of 4D geometry)
Yes, you are right that we are far from god's algorithm and perhaps we
always will be. You are also right that this is because the solutions we
use are designed specifically for human use. I did not mean to suggest
that we are getting close to the limits of what is possible; just that
it feels as if we're beginning to feel the limits of these particular
methods. Will we be able to find new, more powerful methods that humans
can still apply? This is a very interesting question though I doubt that
dimensionality has much to do with it.
The July 2008 issue of Scientific American contained an article with a
lovely sidebar titled "Puzzle Tactics" that teaches you how to find a
solution to any twisty puzzle.
(http://www.scientificamerican.com/article.cfm?id=how-to-solve-the-rubiks-cube)
It clarified the way that all these solutions are generated and helped
me master the Megaminx.
Assuming it is true that everyone uses the same high-level approach to
generate solution methods to any twisty puzzle, the real open question
in my mind is whether there are any *other* approaches that can result
in more efficient solution methods. I expect the best wisdom in this
area will be found in the speed-cubing community since they are all
about efficiency. Remi, I think of you as the representative of that
community. What do you think?
-Melinda
Andrey wrote:
> Melinda, Remi, thank you!
> I haven't try 2^4 yet. But it seems to be much more 4D than large cubes - so it will be interesting to do something with it )))
> As for real difficulties of n^4, I think that we are very far from it. All human algorithms are based on some restricitions of possible actions - so we are closing short ways and select long but predictable. I tried to close as few ways as possible - but to have small enough set of resulting positions. I'm sure that there better ways (that use full power of 4D geometry)
Andrey, that makes you sub-1000 for 4^4 and sub-2000 for 5^4, an impressive=
set of records!
It will be interesting to see new methods developed here specifically desig=
ned for 4D hypercubes, and how efficient they can be made to be. It has cr=
ossed my mind that one of the methods used by computers to solve efficientl=
y is the Thistlethwaite algorithm, which has been adapted slightly for huma=
ns (Ryan Heise has a tutorial if you don't know it). I wonder what move co=
unts could be achieved applying it to 4D, though since the last section of =
the solve relies on half-turns the current MC4D interface isn't ideal for i=
t. If 180 degree face turns only counted as one move, then the method coul=
d be interesting, though probably very difficult to apply in 4D. Also, doe=
s anyone have any idea what God's number is for any 4D cubes? Since it has=
n't been pinned down for the 3x3x3, I doubt it will be known any time soon!
I am currently working on fine-tuning my 3^4 method, and will be revisiting=
the 4^4 and 5^4 when I have finished that. Hopefully Melinda is correct: =
like Remi I'm interested in speedcubing so hopefully that will help a littl=
e (I wonder how many speedcubers we have here?).
Now that Andrey has taken on the 4D cubes so dramatically, I have a feeling=
that several people (Roice, Remi, I'm looking at you here) will be having =
another go at the top spots, and we may get some records changing hands som=
e more. Could be exciting!
Well done Andrey, keep up the good work!
Matt
P.S. Remi mentioned different 'steering' for the 2^4 in the new MC4D, did I=
miss something?
--- In 4D_Cubing@yahoogroups.com, Melinda Green
>
> Yes, you are right that we are far from god's algorithm and perhaps we=20
> always will be. You are also right that this is because the solutions we=
=20
> use are designed specifically for human use. I did not mean to suggest=20
> that we are getting close to the limits of what is possible; just that=20
> it feels as if we're beginning to feel the limits of these particular=20
> methods. Will we be able to find new, more powerful methods that humans=20
> can still apply? This is a very interesting question though I doubt that=
=20
> dimensionality has much to do with it.
>=20
> The July 2008 issue of Scientific American contained an article with a=20
> lovely sidebar titled "Puzzle Tactics" that teaches you how to find a=20
> solution to any twisty puzzle.=20
> (http://www.scientificamerican.com/article.cfm?id=3Dhow-to-solve-the-rubi=
ks-cube)=20
> It clarified the way that all these solutions are generated and helped=20
> me master the Megaminx.
>=20
> Assuming it is true that everyone uses the same high-level approach to=20
> generate solution methods to any twisty puzzle, the real open question=20
> in my mind is whether there are any *other* approaches that can result=20
> in more efficient solution methods. I expect the best wisdom in this=20
> area will be found in the speed-cubing community since they are all=20
> about efficiency. Remi, I think of you as the representative of that=20
> community. What do you think?
>=20
> -Melinda
>=20
> Andrey wrote:
> > Melinda, Remi, thank you!
> > I haven't try 2^4 yet. But it seems to be much more 4D than large cub=
es - so it will be interesting to do something with it )))
> > As for real difficulties of n^4, I think that we are very far from it=
. All human algorithms are based on some restricitions of possible actions =
- so we are closing short ways and select long but predictable. I tried to =
close as few ways as possible - but to have small enough set of resulting p=
ositions. I'm sure that there better ways (that use full power of 4D geomet=
ry)
>
I use my own version of Thistlethwaite's-like algorithm for 3^3. Actually, =
I know only the common idea of the original version, so all stages were dev=
eloped from the begining. Sequence of operations for 3^3 looks like this:
1) orientation of edges
[single-turns of 2 faces are forbidden]
2a) orientation of corners
2b) first rearrangement of edges
[single-turns of 4 faces are forbidden]
3a) first rearrangement of corners
3b) second rearrangement of corners (alignment of tetrahedrons)
3c) second rearrangement of edges
[all single-turns are forbidden or enabled in some "operations"]
4a) third rearrangement of corners (usually it's 1-2 twists)
4b) third rearrangement of edges
[all turns of two faces are forbidden, cube is reduced to 3^2]
5) final stage - not more than 8 turns
You can see how it works in my log files - last stages of all three 4D cube=
s are 3^3. Yes, it's not very efficient with its double-twists (but I can t=
ake back something by usage of middle layers - that is counted as 2 turns i=
n classic algorithm).
I tried to develop something like that for 3^4. First stages were beautif=
ul - to limit twists of 4 sides to D4 (group of 4-prism) and orient 2C cubi=
es, then limit all sides to D4 (reduce cube to puzzle of two tori), but the=
n... Looks like I need to solve more puzzles to find proper operations for =
them. When I counted steps for some way of solving, I got much more than 30=
0 and decided to find another way of group reduction.
God's number for 3^4 is more than 55 (log(N)/log(23*7)), and I think that=
is around 60-70.
Good luck!
Andrey
--0-51587716-1275992774=:3292
Content-Type: text/plain; charset=utf-8
Content-Transfer-Encoding: quoted-printable
Hi Andrey,
I don't think that God's number will be that far away from the lower bound.=
Just look at the 3^3. The lower bound by counting arguements ist 18 and Go=
d's number seems to be 20. Therefore I think God's number should be anywher=
e in between 58-62, but I don't expect it to be as high as 70.=20
btw: you set some nice records in the meantime. Congratulations!
Happy Hypercubing, Klaus
God's number for 3^4 is more than 55 (log(N)/log(23*7)), and I think that i=
s around 60-70.
Good luck!
Andrey
=20
=20=20=20=20=20=20
--0-51587716-1275992774=:3292
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: quoted-printable
God's number for 3^4 is more than 55 (log(N)/log(23*7)), and I think that=
is around 60-70.
Good luck!
Andrey